\(\int \frac {\arccos (a x)^2}{x^3} \, dx\) [19]

Optimal result

Integrand size = 10, antiderivative size = 43 \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{x}-\frac {\arccos (a x)^2}{2 x^2}+a^2 \log (x) \]

[Out]

-1/2*arccos(a*x)^2/x^2+a^2*ln(x)+a*arccos(a*x)*(-a^2*x^2+1)^(1/2)/x

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4724, 4772, 29} \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{x}+a^2 \log (x)-\frac {\arccos (a x)^2}{2 x^2} \]

[In]

Int[ArcCos[a*x]^2/x^3,x]

[Out]

(a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/x - ArcCos[a*x]^2/(2*x^2) + a^2*Log[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4772

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(d*f*(m + 1))), x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d
+ e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^2}{2 x^2}-a \int \frac {\arccos (a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{x}-\frac {\arccos (a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx \\ & = \frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{x}-\frac {\arccos (a x)^2}{2 x^2}+a^2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{x}-\frac {\arccos (a x)^2}{2 x^2}+a^2 \log (x) \]

[In]

Integrate[ArcCos[a*x]^2/x^3,x]

[Out]

(a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/x - ArcCos[a*x]^2/(2*x^2) + a^2*Log[x]

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arccos \left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x a}+\ln \left (a x \right )\right )\)	\(47\)
default	\(a^{2} \left (-\frac {\arccos \left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x a}+\ln \left (a x \right )\right )\)	\(47\)

[In]

int(arccos(a*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2*arccos(a*x)^2/a^2/x^2+arccos(a*x)/x/a*(-a^2*x^2+1)^(1/2)+ln(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=\frac {2 \, a^{2} x^{2} \log \left (x\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} a x \arccos \left (a x\right ) - \arccos \left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccos(a*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(2*a^2*x^2*log(x) + 2*sqrt(-a^2*x^2 + 1)*a*x*arccos(a*x) - arccos(a*x)^2)/x^2

Sympy [F]

\[ \int \frac {\arccos (a x)^2}{x^3} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x^{3}}\, dx \]

[In]

integrate(acos(a*x)**2/x**3,x)

[Out]

Integral(acos(a*x)**2/x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=a^{2} \log \left (x\right ) + \frac {\sqrt {-a^{2} x^{2} + 1} a \arccos \left (a x\right )}{x} - \frac {\arccos \left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccos(a*x)^2/x^3,x, algorithm="maxima")

[Out]

a^2*log(x) + sqrt(-a^2*x^2 + 1)*a*arccos(a*x)/x - 1/2*arccos(a*x)^2/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (39) = 78\).

Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.91 \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=-\frac {1}{2} \, {\left ({\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \arccos \left (a x\right ) - 2 \, a \log \left ({\left | x \right |}\right )\right )} a - \frac {\arccos \left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccos(a*x)^2/x^3,x, algorithm="giac")

[Out]

-1/2*((a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*arccos(a*x
) - 2*a*log(abs(x)))*a - 1/2*arccos(a*x)^2/x^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x^3} \,d x \]

[In]

int(acos(a*x)^2/x^3,x)

[Out]

int(acos(a*x)^2/x^3, x)